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Abstract
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Let
k 1
be an integer and let
G
denote a finite and simple graph with vertex set
V (G). A
signed double Roman
k
-dominating function on
G
is a mapping
f : V (G) { 1, 1, 2, 3}
satisfying the following: (i) if
f (v) = 1
for a vertex v, then v is adjacent either to a vertex w
with
f (w) = 3
or to at least two vertices assigned value
2
under
f
; (ii) if
f (v) = 1, then v has
a neighbor w with
f (w) 2
; and (iii) for every vertex v, u N [v] f (u) k. The weight of a
signed double Roman
k
-dominating function
f
is given by
f (u) u V (G) , and the minimum
possible weight is called the signed double Roman
k
-domination number of
G. In this
paper, we investigate the signed double Roman
k
-domination number in unicyclic graphs,
where lower bounds are established for
k {1, 2, 3, 4}. Moreover, a characterization of
extremal unicyclic graphs reaching these bounds when
k {1, 2}
is provided.
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